Equivariant triple intersections
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 3, pp. 601-643.

Étant donné un nœud K dans une sphère d’homologie rationnelle M, et le revêtement infini cyclique standard X ˜ de (M,K), on définit un invariant des triplets de courbes dans X ˜, via des intersections triples équivariantes de surfaces. On montre que cet invariant fournit une application φ sur 𝔄 3 , où 𝔄 est le module d’Alexander de (M,K), et que la classe d’isomorphisme de φ est un invariant de la paire (M,K). Pour un module de Blanchfield (𝔄,𝔟) fixé, on considère les paires (M,K) dont le module de Blanchfield est isomorphe à (𝔄,𝔟), équippées d’un marquage, c’est-à-dire d’un isomorphisme fixé de (𝔄,𝔟) vers le module de Blanchfield de (M,K). Dans ce cadre, on calcule la variation de φ sous l’effet d’une chirurgie borroméenne nulle, et on décrit l’ensemble de toutes les applications φ. Enfin, on montre que l’application φ est un invariant de type fini de degré 1 des paires marquées (M,K) par rapport aux chirurgies LP nulles, et on détermine l’espace de tous les invariants de degré 1 à valeurs rationnelles des paires marquées (M,K).

Given a null-homologous knot K in a rational homology 3-sphere M, and the standard infinite cyclic covering X ˜ of (M,K), we define an invariant of triples of curves in X ˜ by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map φ on 𝔄 3 , where 𝔄 is the Alexander module of (M,K), and that the isomorphism class of φ is an invariant of the pair (M,K). For a fixed Blanchfield module (𝔄,𝔟), we consider pairs (M,K) whose Blanchfield modules are isomorphic to (𝔄,𝔟) equipped with a marking, i.e. a fixed isomorphism from (𝔄,𝔟) to the Blanchfield module of (M,K). In this setting, we compute the variation of φ under null Borromean surgeries and we describe the set of all maps φ. Finally, we prove that the map φ is a finite type invariant of degree 1 of marked pairs (M,K) with respect to null Lagrangian-preserving surgeries, and we determine the space of all degree 1 invariants with rational values of marked pairs (M,K).

Reçu le :
Accepté le :
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DOI : 10.5802/afst.1547
Classification : 57M27, 57M25, 57N65, 57N10
Mots clés : Knot, Homology sphere, Equivariant intersection, Alexander module, Blanchfield form, Borromean surgery, Null-move, Lagrangian-preserving surgery, Finite type invariant.
Moussard, Delphine 1

1 Institut de Mathématiques de Bourgogne, 9 avenue Alain Savary, 21000 Dijon, France
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Moussard, Delphine. Equivariant triple intersections. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 3, pp. 601-643. doi : 10.5802/afst.1547. http://archive.numdam.org/articles/10.5802/afst.1547/

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